Application of unitary representations of the Lorentz group to an analysis of the dynamic symmetry of one-dimensional quantum systems

Abstract

In the present work, it is demonstrated that for some one-dimensional quantum systems it is convenient to seek for the finite-dimensional Lie algebras of shift operators of eigenvectors in the framing algebra above the Heisenberg algebra with four generatrices. Within the framework of this approach, the dynamic symmetry of the Schrodinger equations with potentials 1/cosh2(×), 1/sinh2(×), and exp(−2×), which can be associated with the Lorentz group, is considered. The representations are analyzed and compared with values of the potential. Conditions at which the potential 1/sinh2(×) is reflectionless are determined.